A368794 - OEIS

%I #13 Feb 13 2024 08:14:32

%S 0,1,2,11,76,685,7534,97943,1469144,24975449,474533530,9965204131,

%T 229199695012,5729992375301,154709794133126,4486584029860655,

%U 139084104925680304,4589775462547450033,160642141189160751154,5943759223998947792699,231806609735958963915260

%N a(n) = (2*n-1)!! * Sum_{k=1..n} (-1)^(k-1)/(2*k-1)!!.

%F a(0) = 0; a(n) = (2*n-1)*a(n-1) + (-1)^(n-1).

%F From _Peter Bala_, Feb 10 2024: (Start)

%F a(n) = (2*n - 2)*a(n-1) + (2*n - 3)*a(n-2) with a(0) = 0 and a(1) = 1.

%F The double factorial numbers (2*n-1)!! = A001147(n) satisfy the same recurrence, leading to the generalized continued fraction expansion Limit_{n -> oo} a(n)/(2*n-1)!! = Sum_{k >= 1} (-1)^(k-1)/(2*k-1)!! = 0.7247784590... = 1 - 1/(3 + 3/(4 + 5/(6 + 7/(8 + 9/(10 + ... )))))). (End)

%o (PARI) a001147(n) = prod(k=1, n, 2*k-1);

%o a(n) = a001147(n)*sum(k=1, n, (-1)^(k-1)/a001147(k));

%Y Cf. A001147, A286286, A306858.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jan 06 2024